Talk:List of functions
How about including the forms of these functions, for example \(a\{\{1\}\}b\) for expansion? -- I want more 08:42, March 19, 2013 (UTC) According to Bowers, L means all types of arrays up to but not including legion space. It's like how ordinals are sets of all ordinals below themselves. FB100Z • talk • 20:27, March 19, 2013 (UTC) Aarex's second function is not well-defined (it recurses to larger numbers). If corrected it will be about \(\omega^2\) past the Xi function. Deedlit11 (talk) 22:30, March 21, 2013 (UTC) The growth rate of Laver table, Friedman's finite games , Loader.c and fusible function are unknown. But why the former three are in the "From \(\Gamma_0\) to \(\omega_1^{CK}\)" section but the last one is in the "other" section? hyp$hyp?cos&cos (talk) 04:23, January 4, 2014 (UTC) : Fusible function has not been yet proven to be over \(\varepsilon_0\) rate growth, so we don't really know which section it belongs to. For Loader's and Friedman's function we know with certainty they fit their category. I'll move Laver table, because we don't even know if it's well defined. LittlePeng9 (talk) 08:59, January 4, 2014 (UTC) Picture Can someone make an scrolling bar for the large picture I tried to add to the article? Wythagoras (talk) 07:06, February 22, 2014 (UTC) \(\Xi\) function Last 4 edits make it wrong. It has growth rate \(\alpha\mapsto\omega^{CK}_\alpha\). Why someone thinks it's \(\omega^{CK}_\omega\)? {hyp/^,cos} (talk) 02:36, August 28, 2014 (UTC) :Well, to be precise, we don't know if it reaches even \(\omega_2^\text{CK}\) (see here). LittlePeng9 (talk) 06:31, August 28, 2014 (UTC) Very important! We got a recent discussion on IRC about this page. Here you can read the whole discussion: http://pastebin.com/141W721a We thought that it would be better to make a special article about BEAF ill-defined numbers and remove them from this page (if you don't know it, the community has found out that BEAF is ill-defined beyond tetrationnal level). The scale will be changed: everything after "Tetrational array notation level" (which includes "Higher operations level", "Legiattic array notation level", and "Beyond Legiattic arrays notation level") will be put in "Beyond tetrational arrays notation level", and numbers beyond Triakulus (included Triakulus itself) these has been defined using BEAF beyond tetrational level will be put in a new section or a new article (more likely a new article) Fluoroantimonic Acid (talk) 19:42, July 9, 2015 (UTC) Good idea? Yes, do the edit No, don't do anything :well, beyond tetrational arrays is kind of a huge range. Cookiefonster (talk) 19:47, July 9, 2015 (UTC) :we shall find an alternative. if you have a proposition say it Fluoroantimonic Acid (talk) 19:52, July 9, 2015 (UTC) Section titles I have retitled the sections so they refer to arithemetic theories and not FGH. Although the sections are rough, I think it's better to compare with well-defined boundaries rather than ill-defined ones. I don't want to piss anyone off, so no other FGH comparisons were removed. -- ve 19:56, August 31, 2015 (UTC) rename i suggest renaming to "List of functions" -- ve 09:11, September 13, 2015 (UTC) :No response on this, so I'm just going to move. -- ve 20:21, October 3, 2015 (UTC) :How was it called before?Boboris02 (talk) 14:08, November 5, 2016 (UTC) ::"List of googological functions" -- ☁ I want more ⛅ 14:13, November 5, 2016 (UTC) Theory or Growth Rate The need of a stronger theory to prove total does not imply a higher growth rate. This leads to some problems. An example is the fusible margin function; it is proven total in ZFC, which is a strong theory, but its growth rate is unknown and might be very low, even possible lower than all functions listed in section "ZFC set theory". A similar example is the Laver's q function; it is proven total in a stronger theory than ZFC, but with unknown growth rate. Another problem is that Placid platypus function and Weary wombat function are uncomputable but with growth rates bounded by some computable functions. Even though a function needs strong theory to prove total or is uncomputable, it is possible eventually dominated by some functions proven total in some weak theory or computable. In such cases, "ordered by theory" and "ordered by growth rate" give different lists. What's more, the theories used in fusible margin function and Laver's q function might not be optimal. There might be weaker theories to prove them total. Unsolved growth rates and theories should be placed in "other". {hyp/^,cos} (talk) 08:11, January 8, 2020 (UTC) : If you stated that we should create two lists on growth rates and theories, then I would agree with it. If you want to put functions with unknown growth rates which are not necessarily well-defined in ZFC set thoery into "Other" category, then should we put all functions whose growth rates are effectively estimated in terms of ordinals? Then maybe almost all functions should be put into that category. Or is there any reason why SAN and BMS are suitable for ZFC? : Anyway, I clarified the issue. Then nobody who carefully reads the article will misunderstand the list. Are you happy now? : p-adic 09:23, January 8, 2020 (UTC) The use of theories in the list is not suitable. Instead, I suggest using "these functions eventually dominate all functions provably recursive in T" where T is a theory. It has two advantages: #In this way, stronger theory means higher growth rate. #It naturally marks milestone ordinals (at their PTO), and the estimations in terms of ordinals are meaningful. But this site lacks an introduction for a theory with limit growth rate \(\omega^\omega\) (or PTO \(\omega^{\omega^\omega}\)). {hyp/^,cos} (talk) 11:50, January 8, 2020 (UTC) : I think that your idea is reasonable. But are you planning to put many functions such as SAN, BMS, and many other functions such that the eventual domination has never been verified into "Other" category? I note that "F is of growth rate α in FGH with respect to a certain system of fundamental sequences" does not imply "F eventually dominates all computable functions recursively total in T with PTO ≦ α". Therefore there are few functions which we can categorise in that way. : Although I do not know well about weak arithmetics, PRA is known to be of PTO ω^ω. : p-adic 12:02, January 8, 2020 (UTC) ::What is the case that "f has growth rate α" but f does not "eventually dominates all functions provably recursive in T with PTO ≦ α"? Is it the usage of some fundamental sequences? {hyp/^,cos} (talk) 12:25, January 8, 2020 (UTC) ::: Right. For example, ω^ω in FGH with respect to the system of fundamental sequences is weaker than f_{ω+1}. You can find related arguments here. Since FGH has nothing to do with proof theory, we do not have clear relation between ordinals in FGH and PTOs even if you choose "natural" or "canonical" systems of fundamental sequences. (I note that there are several study on the relation for specific fundamental sequences.) ::: p-adic 13:17, January 8, 2020 (UTC)